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A194725 Number of 5-ary words either empty or beginning with the first character of the alphabet, that can be built by inserting n doublets into the initially empty word. 4
1, 1, 9, 97, 1145, 14289, 185193, 2467137, 33563481, 464221105, 6507351113, 92236247841, 1319640776249, 19031570387857, 276368559434025, 4037555902072065, 59299855337012505, 875056238174271345, 12967283824008178185, 192889769468751321825, 2879117809973276680185 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..200

C. Kassel and C. Reutenauer, Algebraicity of the zeta function associated to a matrix over a free group algebra, arXiv preprint arXiv:1303.3481, 2013

FORMULA

G.f.: 4/5 + 8/(5*(3+5*sqrt(1-16*x))).

a(0) = 1, a(n) = 1/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*4^j for n>0.

a(n) ~ 2^(4*n+2)/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013

Recurrence: n*a(n) = (41*n-24)*a(n-1) - 200*(2*n-3)*a(n-2). - Vaclav Kotesovec, Aug 13 2013

EXAMPLE

a(2) = 9: aaaa, aabb, aacc, aadd, aaee, abba, acca, adda, aeea (with 5-ary alphabet {a,b,c,d,e}).

MAPLE

a:= n-> `if`(n=0, 1, add(binomial(2*n, j) *(n-j) *4^j, j=0..n-1) /n):

seq(a(n), n=0..20);

# second Maple program

a:= proc(n) a(n):= `if`(n<3, [1, 1, 9][n+1],

       ((41*n-24)*a(n-1) +(600-400*n)*a(n-2))/n)

    end:

seq(a(n), n=0..20);  # Alois P. Heinz, Oct 30 2012

MATHEMATICA

FullSimplify[Flatten[{1, Table[4^(2*n+1)*(1/2 (2*n-1))! Hypergeometric2F1[1, 1/2+n, 2+n, 16/25]/(25*Sqrt[Pi]*(n+1)!), {n, 1, 20}]}]] (* Vaclav Kotesovec, Aug 13 2013 *)

CROSSREFS

Column k=5 of A183134.

Sequence in context: A241771 A180675 A083077 * A218500 A293986 A123821

Adjacent sequences:  A194722 A194723 A194724 * A194726 A194727 A194728

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Sep 02 2011

STATUS

approved

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Last modified February 25 04:05 EST 2018. Contains 299630 sequences. (Running on oeis4.)